Sec. 50.2 



CALCULATION OF WAVEMAKING RESISTANCE 



209 



oid," signifying a ship surface over which the 

 water might ghde easily [INA, 18G4, pp. 327-328]. 

 Of these surfaces he said: 



"The co-efRcients of those Lissoneoids which are of 

 proportions available for practical use (the length being 

 four times the breadth, and upwards) range from three- 

 fifths to two-thirds, and differ but Uttle in any case from 

 0.637, which is also the co-efficient of fineness for a curve 

 of sines" [INA, 1864, p. 327). 



The latter corresponded to J. Scott Russell's 

 waterline curve of versed sines, reproduced in 

 diagram A of Fig. 24. G in Volume I. It seems 

 clear that Rankine, as a ship designer of that day 

 as well as a mathematician and scientist of con- 

 siderable renown, did use shapes resembling 

 lissoneoids for the waterlines of some ships. 



Rankine points out, in his 1864 paper, cited 

 earlier in this section, that along a lissoneoid, 

 clear of the stream form, there is a region of 

 maximum positive differential pressure +Ap 

 abreast the leading edge and another abreast the 

 trailing edge, with only one region of maximum 

 negative differential pressure — Ap abreast the 

 middle of the body. Around the neoid or stream 

 form proper there are two points, one on each 

 shoulder, where the Ap drops to its maximum 

 negative numerical value, while at amidships it 

 rises to a somewhat lesser negative numerical 

 value. This means that, around and along a 

 neoid, the pressure changes the sign of its longi- 

 tudinal gradient (rises and falls) six times, 

 whereas along a lissoneoid it changes only four 

 times. This situation is shown schematically in 

 diagram 2 of Fig. 50. A. The significance of these 

 changes was appreciated by Rankine, who felt 

 that the fewer the pressure changes along the 

 side of a ship the better. 



With this background, and some previous woi'k 

 on waves [Phil. Trans., Roy. Soc, 1863], Rankine 

 developed two formulas, essentially similar, for 

 calculating the resistance of a ship having a 

 certain shape of waterline. They are described 

 here at some length, not so much because of their 

 interesting historic value, or of their practical 

 and physical worth, but because of the lines of 

 reasoning by which they were derived. It is of 

 Uttle present importance that the hydrodynamics 

 was faulty or that they proved inadequate for 

 design purposes but it is important that the 

 derivation was based on considerations of hydro- 

 dynamics rather than on the intensely practical 

 nautical knowledge of that day. 



Rankine considered, first, that the length L of 



the ship was matched by the length L„. of a 

 trochoidal wave traveling at the same speed, 

 with two adjacent crests opposite the bow and 

 stern, and with its trough amidships. Second, he 

 assumed a hypothetical barge-shaped ship with 

 a given constant beam and a vertically curved 

 bottom, shaped to a nicety so that it fitted exactly 

 into the trough of the trochoidal wave and rested 

 everywhere upon the wave surface, like a sailor 

 relaxing in a hammock. He took for granted, as 

 did everyone else at that time, that the friction 

 resistance along the ship's side varied as the 

 square of the speed. Further, that the friction 

 resistance along the under side of the barge- 

 shaped ship was balanced by a forward thrust or 

 propelling force which, exerted by the water under 

 the run, just counteracted the friction effects 

 upon the whole barge. This procedure appeared 

 to be exceedingly clever, because it obviated the 

 necessity of estimating or computing those effects. 

 Rankine then transformed the curved-bottom 

 barge into a ship with a mean girth equal to the 

 barge's beam, and with a run which was charac- 

 terized by a waterline having the shape of a 

 trochoid. Utilizing a few more transformations 

 that read like a story of alchemy, Rankine 

 evolved a formula for the total resistance of the 

 ship. Rewritten in the notation of this book, his 

 formula states that the total ship 



Resistance = Cf;^- V'L{ . ,, I 

 2g Vgirth / 



(50.i) 



■ [1 + 4 sin^ in + sin* in] 



■ [1 -F 4 sin' ia + sin* in] 



where Ir is the maximum slope of the trochoidal 

 waterline in the run and V is the ship speed. 

 The third term in the brackets, sin* Ir , may 

 become small enough to be neglected in compari- 

 son with the others. Based on the use of English 

 units of measurement, the friction coefficient 

 Cp , according to the knowledge of that time, 

 was about 0.0036 for the clean, painted surfaces 

 of iron ships ["On the Computation of the 

 Probable Engme-Power and Speed of Proposed 

 Ships," INA, 1864, pp. 316-333]. 



The terms to the left of the brackets, combined 

 with the first term within them, namely unity, 

 gave the friction resistance only. The pressure 

 resistance was added by taking account of the 



